Fractional infinity Laplacian with obstacle

This paper deals with the obstacle problem for the fractional infinity Laplacian with nonhomogeneous term f(u), where f : ℝ+ ↦ ℝ+ : {L[ u ] = f(u) in { u > 0} u ≥ 0 in Ω u = g on ∂ Ω with L [u ](x) − supy∈Ω,y≠x u (y) − u(x)/| y−x |α + infy∈Ω, y ≠ x u(y) − u (x)/| y − x |α, 0 < α < 1. Under the assumptions that f is a continuous and monotone function and that the boundary datum g is in C0,β (∂Ω) for some 0 < ß < α, we prove existence of a solution u to this problem. Moreover, this solution u is β—Hölderian on ̅Ω. Our proof is based on an approximation of f by an appropriate sequence of functions fɛ where we prove using Perron's method the existence of solutions uɛ, for every ɛ > 0. Then, we show some uniform Hölder estimates on uɛ that guarantee that uɛ → u where this limit function u turns out to be a solution to our obstacle problem.